Linking Early and Late Type Galaxies to their Dark Matter Haloes

Transcription

1 Mon. Not. R. Astron. Soc., (23) Linking Early and Late Type Galaxies to their Dark Matter Haloes Frank C. van den Bosch 1, Xiaohu Yang 1,2,andH.J.Mo 1 1 Max-Planck-Institut für Astrophysik, Karl Schwarzschild Str. 1, Postfach 1317, Garching, Germany 2 Center for Astrophysics, University of Science and Technology of China, Hefei, Anhui 2326, China Accepted... Received...; in original form... ABSTRACT Using data from the 2 degree Field Galaxy Redshift Survey (2dFGRS) we compute the conditional luminosity functions (CLFs) of early- and late-type galaxies. These functions give the average number of galaxies with luminosities in the range L ± dl/2 that reside in a halo of mass M, and are a powerful statistical tool to link the distribution of galaxies to that of dark matter haloes. Although some amount of degeneracy remains, the CLFs are well constrained. They indicate that the average mass-to-light ratios of dark matter haloes have a minimum of 1h (M/L) around a halo mass of h 1 M. Towards lower masses M/L increases rapidly, and matching the faint-end slope of the observed luminosity function (LF) requires that haloes with M<1 1 h 1 M are virtually devoid of galaxies. At the high mass end, the observed clustering properties of galaxies require that clusters have b J -band mass-to-light ratios in the range 5 1 h (M/L). Finally, the fact that early-type galaxies are more strongly clustered than late-type galaxies requires that the fraction of late-type galaxies is a strongly declining function of halo mass. We compute two-point correlation functions as function of both luminosity and galaxy type. The agreement with observations, in terms of normalization and power-law slope, is remarkably good. When including predictions for the correlation functions of faint galaxies we find a weak (strong) luminosity dependence for the late (early) type galaxies. We also investigate the inferred halo occupation numbers. Late-type and faint galaxies reveal a shallower N (M) than bright, early-type galaxies, which explains why N (M) transforms from a single power-law for bright galaxies to a more complicated form when fainter galaxies are included. Finally we compare our CLFs with predictions from several semi-analytical models for galaxy formation. As long as these models accurately fit the 2dFGRS luminosity function the agreement with our predictions is remarkably good. This indicates that the technique used here has recovered a statistical description of how galaxies populate dark matter haloes which is not only in perfect agreement with the data, but which in addition fits nicely within the standard framework for galaxy formation. Key words: galaxies: formation galaxies: clusters large-scale structures: cosmology: theory dark matter 1 INTRODUCTION According to the current paradigm galaxies form and reside inside extended cold dark matter (CDM) haloes. One of the ultimate challenges in astrophysics is to obtain a detailed understanding of how galaxies with different properties occupy haloes of different masses. This link between galaxies and dark matter haloes is an imprint of various complicated physical process related to galaxy formation such as gas cool- vdbosch@mpa-garching.mpg.de ing, star formation, merging, tidal stripping and heating, and a variety of feedback processes. One method to investigate the galaxy-dark matter connection is therefore to consider ab initio models for galaxy formation, using either numerical simulations (e.g., Katz, Weinberg & Hernquist 1996; Fardal et al. 21; Pearce et al. 2; Kay et al. 22) and/or semi-analytical models (e.g., White & Rees 1978; Kauffmann, White & Guiderdoni 1993; Somerville & Primack 1999; Kauffmann et al. 1999; Cole et al. 2; Benson et al. 22; van den Bosch 22). A downside of this approach, however, is that phenomenological descriptions have to be used to describe a variety

2 2 van den Bosch, Yang & Mo of poorly understood physical processes. Consequently, an alternative method has been developed which completely sidesteps the uncertainties related to how galaxies form. This method tries to infer the link between galaxies and dark matter haloes directly from the observed clustering properties of galaxies. Since haloes of different mass and galaxies of different luminosity and type are all clustered differently, there is only a limited amount of possibilities by which one can distribute galaxies over dark matter haloes such that their clustering properties are consistent with observations. The backbone of this approach is the so-called halo model, which views the evolved, non-linear dark matter distribution in terms of its halo building blocks: on strongly non-linear scales the dark matter distribution is given by the actual density distributions of the virialized haloes, while on larger, close-to-linear scales, the dark matter distribution is given by the distribution of virialized haloes (see Cooray & Sheth 22 for a detailed review). This halo model has become more and more accurate due to the fact that detailed analytical descriptions for the structure and clustering of dark matter haloes have become available (e.g., Navarro, Frenk & White 1997; Moore et al. 1998; Bullock et al. 21; Mo & White 1996, 22; Power et al. 22). The halo model can be naturally extended to address the bias of galaxies by introducing a model for the halo occupation numbers, N(M), which describes how many galaxies on average (with luminosities L>L min) occupy a halo of mass M. Numerous studies in the past have used these halo occupation number models to investigate how changes in N(M) impact on several statistical properties of the galaxy distribution, such as the real-space two- and threepoint correlation functions, the power spectrum and bispectrum of galaxies, the galaxy-mass cross correlation function, the pair-wise velocity dispersions, etc. (Seljak 2; Scoccimarro et al. 21; White 21; Berlind & Weinberg 22; Scranton 22a; Kang et al. 22; Marinoni & Hudson 22; Kochanek et al. 22). In addition, several studies have confronted these models with data to put constraints on N(M) (Jing, Mo & Börner 1998; Peacock & Smith 2; Marinoni & Hudson 22; Kochanek et al. 22; Jing, Börner & Suto 22; Bullock, Wechsler & Somerville 22). With large galaxy redshift surveys becoming available, such as the Two-Degree Field Galaxy Redshift Survey (2dFGRS; see Colless et al. 21) and the Sloan Digital Sky Survey (SDSS, see York et al. 2), these models can now be confronted with statistical data of unprecedented quality to obtain stringent constraints on the halo occupation numbers, and therewith on both cosmological parameters and galaxy formation models. In a recent paper, Yang, Mo & van den Bosch (22; hereafter Paper 1) have taken this approach one step further by considering the derivate of N(M) with respect to L min. In particular, they introduced the conditional luminosity function (hereafter CLF) Φ(L M)dL, which gives the average number of galaxies with luminosities in the range L ± dl/2 that reside in haloes of mass M. The advantage of the CLF over the halo occupation function N(M) is that it allows one to address the clustering properties of galaxies as function of luminosity. In addition,the CLF yields a direct link between the halo mass function n(m)dm, which gives the number of dark matter haloes per comoving volume with masses in the range M ± dm/2, and the galaxy luminosity function (hereafter LF) Φ(L)dL, which gives the number of galaxies per comoving volume with luminosities in the range L ± dl/2, according to Φ(L) = Φ(L M) n(m)dm. (1) Therefore, Φ(L M) is not only constrained by the clustering properties of galaxies, as is the case with N(M), but also by the observed galaxy luminosity function. Furthermore, knowledge of Φ(L M)dL allows one to compute the average total luminosity of galaxies in a halo of mass M, L (M) = Φ(L M) L dl (2) and thus the average mass-to-light ratio as function of halo mass. This M/L (M) yields important constraints on galaxy formation models, as it is a direct measure of the halo mass dependence of the galaxy formation efficiency. In Paper 1 we focussed on the cosmology dependence of the CLF. In this paper, we use data from the 2dFGRS to compute the CLFs of both early and late-type galaxies and compare our results with several semi-analytical models for galaxy formation. It is well known that galaxies of different morphological types have different luminosity functions and different clustering properties. For example, early-type galaxies have higher characteristic luminosities (e.g., Efstathiou, Ellis & Peterson 1988; Loveday et al. 1992) and are more strongly clustered (e.g., Willmer, Da Costa & Pellegrini 1998; Zehavi et al. 22) than late-type galaxies. The CLFs presented here allow an interpretation of these morphological dependencies in terms of halo occupation numbers, while our comparison with the semi-analytical models links their physical origin to the framework of galaxy formation models. Throughout this paper we define M to be the halo mass inside the radius R 18 inside of which the average density is 18 times the cosmic mean density, and h is the Hubble constant in units of 1 km s 1 Mpc 1. 2 OBSERVATIONAL CONSTRAINTS In paper 1 we used data from the 2dFGRS to constrain the CLF of the entire galaxy population. This data set included over 115 galaxies with 17. <b J < 19.2 andz<.25. In this paper we are interested in the CLFs of the earlyand late-type galaxies, for which we need separate LFs and separate measurements of the luminosity dependence of the clustering properties. Madgwick et al. (22) used a principal component analysis of galaxy spectra taken from the 2dFGRS to obtain a spectral classification scheme. They introduced the parameter η, a linear combination of the two most significant principal components, as a galaxy type classification measure. As shown by Madgwick et al. (22), η follows a bimodal distribution and can be interpreted as a measure for the current star formation rate in each galaxy. Furthermore η is well correlated with morphological type (Madgwick 22). In what follows we adopt the classification suggested by Madgwick et al. and classify galaxies with η< 1.4 as early-types and galaxies with η 1.4 as late-types. We caution the reader that despite the good correlation between

3 Linking Early and Late Type Galaxies to their Dark Matter Haloes 3 Figure 1. The data used to constrain the models. The left panel plots the LFs of the early-type galaxies (starred symbols), the late-type galaxies (open triangles), and the combined sample of late- plus early-type galaxies (open circles). For clarity, no errorbars are shown (but see Figure 2 below). The dotted vertical lines indicate the luminosity range over which measurement of clustering correlation lengths are available. Right panel plots these galaxy-galaxy correlation lengths as function of luminosity (same symbols); Again, for clarity no errorbars are shown (but see Figure 2). η and morphological type, there is not a one-to-one correspondence between our description of early- and late-type galaxies and those obtained using a more morphological criterion. for the parameters), (ii) early- (late-) type galaxies dominate the total LF at the bright (faint) end, and (iii) the LF of the early-type galaxies reveals a remarkable, though only marginally significant, upturn at the faint-end. 2.1 Luminosity Functions In order to constrain the CLFs of early- and late-type galaxies we adopt the LFs in the photometric b J band computed by Madgwick et al. (22) from the 2dFGRS. This sample is restricted to the redshift range z.15 and contains unique galaxies with accurate type-classification based on η. About 36 percent of these galaxies have η< 1.4 andmake up what we refer to as the sample of early-type galaxies; the remainder makes up the late-type galaxies. In what follows, we shall refer to the entire sample (both late- and early-type galaxies) as the combined sample. The LFs obtained from this sample of galaxies have been corrected for completeness effects, and a self-consistent method for k-corrections, based on the observed 2dF spectra, has been applied. A cosmology with Ω =.3 and Ω Λ =.7 has been adopted, which corresponds to the same cosmological parameters as we adopt throughout this paper. The left panel of Figure 1 plots the LFs for the combined sample (circles), as well as its contributions from latetype (triangles) and early-type (starred symbols) galaxies. In their paper, Madgwick et al. (22) subdivided the latetype galaxies in three sub-types. The late-type LF used here is computed by summing the LFs of these three subtypes, and by adding the errors in quadrature. For clarity, no errorbars have been plotted, but these can be seen in Figure 2 below. The main characteristics of these LFs are that (i) the LFs of the combined and late-type samples are well fit by a Schechter (1976) function (see Madgwick et al. (22) 2.2 Correlation Lengths It is straightforward to see that there is an infinite set of CLFs that, given a halo mass function, can reproduce the observed LF Φ obs (L) of galaxies. For example, all CLFs given by Φ(L M) = [Φ obs (L)/n(M )]δ k (M M ), where δ k (x) is the Kronecker delta function and M is an arbitrary mass, yield exactly the same LF, namely Φ obs (L) (see equation [1]). Therefore, in order to constrain the CLF, additional constraints are required. Since more massive dark matter haloes are more strongly clustered (e.g., Mo & White 1996), galaxy-galaxy correlation lengths as function of luminosity, r (L), put important constraints on how galaxies of different luminosities have to be distributed over haloes of different masses. Indeed, as shown in Paper 1, a combination of an observed LF with measurements of r (L) allow the CLF to be well constrained. Because of the shear size of the 2dFGRS it is possible to compute the clustering properties of galaxies of different spectral types and luminosities in representative volumelimited samples. Norberg et al. (22) obtained real space galaxy-galaxy two-point correlation functions ξ gg(r) from the same sample of 2dFGRS galaxies as that used by Madgwick et al. (22) for the LFs. For a number of volumelimited subsamples (specified by a range in absolute magnitude) Norberg et al. computed ξ gg(r) (assuming Ω =.3 and Ω Λ =.7) separately for the late-type, the early-type, and the combined samples. Each of these correlation functions is well fit by a simple power-law ξ gg(r) = (r/r ) γ.

4 4 van den Bosch, Yang & Mo Although there is no clear trend of γ with either luminosity or spectral type (see Section 6), the correlation lengths r depend strongly on both luminosity and spectral type. This is shown in the right panel of Figure 1, which plots r (L) for the late-type galaxies (triangles), the early-type galaxies (starred symbols), and the combined sample (circles). For clarity, no errorbars are plotted (but see Figure 2 below). Two trends are apparent: typically r increases with luminosity (i.e., more luminous galaxies are more strongly clustered) and early-type galaxies are more strongly clustered than late-type galaxies of the same luminosity. This type-dependence is particularly strong for the fainter galaxies. Note that the correlation function itself may also evolve with redshift, an effect that has not been corrected for. For the redshift range covered by the 2dFGRS data used here (i.e., z.15), this effect is fairly small, but we nevertheless take it into account in our modeling (see Section 3 below). 3 THEORETICAL BACKGROUND Our main goal in this paper is to use the observed LFs and correlation lengths to constrain the CLFs of the early-type and late-type galaxies. Here we briefly describe how to compute LFs and correlation functions from a given CLF. More details can be found in Paper 1 and references therein. To compute the galaxy LF Φ(L) from the CLF Φ(L M) (equation [1]) one needs the (cosmology-dependent) mass function n(m) of dark matter haloes, which (at z =)is given by n(m)dm = ρ M νf(ν) dlnσ dm. (3) 2 dlnm Here ρ is the mean matter density of the Universe at z =, ν = δ c/σ(m), δ c is the critical overdensity required for collapse at z =,f(ν) is a function of ν to be specified below, and σ(m) is the linear rms mass fluctuation on mass scale M, which is given by the linear power spectrum of density perturbations P (k) as σ 2 (M) = 1 P (k) ŴM(k) 2 k 2 dk, (4) 2π 2 where Ŵ M(k) is the Fourier transform of the smoothing filter on mass scale M. Throughout we adopt the form of f(ν) suggested by Sheth, Mo & Tormen (21): νf(ν) =.644 ( 1+ 1 ) ( ν 2 ν.6 2π ) 1/2 exp ( ν 2 2 with ν =.841 ν. The resulting mass function has been shown to be in excellent agreement with numerical simulations, as long as halo masses are defined as the masses inside a sphere with an average overdensity of about 18 (Sheth & Tormen 1999; Jenkins et al. 21; White 22). Therefore, in what follows we consistently use that definition of halo mass, and we use the CDM power spectrum P (k) ofefstathiou, Bond & White (1992) with a spatial top-hat filter for which Ŵ M(k) = 3 [sin(kr) kr cos(kr)] (6) (kr) 3 ) (5) where the mass M and filter radius R are related according to M =4π ρr 3 /3. In order to compute r (L) from the CLF we proceed as follows. In the halo model it is natural to consider the galaxy-galaxy two-point correlation function, ξ gg(r), to be build up from two parts; a 1-halo term, ξgg 1h (r), which represents the correlation due to pairs of galaxies within the same halo, and a 2-halo term, ξgg 2h (r), describing the contribution due to pairs of galaxies that occupy different haloes. The observed correlation lengths are all well in excess of 3.5h 1 Mpc. At radii this large the contribution from the 1-halo term is negligible and for the purpose of calculating correlation lengths only the 2-halo term is required. As we show in Appendix A, the 2-halo term of ξ gg(r) for galaxies with L 1 <L<L 2 canbewrittenas ξgg 2h (r) =b 2 ξdm(r), 2h (7) with ξdm(r) 2h the 2-halo term of the dark matter mass correlation function at z =, which at scales of the correlation length is well fit by a single power-law ( ) 1.75 ξdm(r) 2h r ξ dm (r) (8) r,dm and n(m) N(M) b(m)dm b = (9) n(m) N(M) dm Here L2 N(M) = Φ(L M)dL (1) L 1 is the mean number of galaxies in the specified luminosity range for haloes of mass M, andb(m) is the bias of dark matter haloes of mass M with respect to the dark matter mass distribution (see Appendix A). Note that the 2-halo term of the galaxy-galaxy correlation function is completely specified by the CLF and does not require knowledge about how galaxies are distributed inside individual dark matter haloes. Since b is scale-independent, we can rewrite equation (7) directly in terms of the correlation length for galaxies as r = b r,dm (11) There is one additional effect, though, that we have to take into account. The correlation functions obtained by Norberg et al. (22) have not been corrected for possible redshift evolution. This implies that the r (L) measurements do not correspond to z =. In fact, since more luminous galaxies can be detected out to higher redshift, the correlation lengths of brighter galaxies correspond to galaxy populations with a higher median redshift. Therefore, in order to compare model and observations in a consistent way, we have to compute the correlation lengths at the characteristic redshift of the sample in consideration. We take this into account by replacing b in equation (11) with b eff (z). Here z is the mean redshift of the galaxies in consideration (taken from Norberg et al. 22) and b eff is the effective bias defined in Appendix B. Since the 2dFGRS sample used here is limited to z.15, this redshift correction is only modest, amounting to no more than a few percent change in r.

5 Linking Early and Late Type Galaxies to their Dark Matter Haloes 5 4 MODELING THE CONDITIONAL LUMINOSITY FUNCTION In Paper 1 we adopted a particular parameterization of the CLF for the entire galaxy population (early plus late type galaxies), and we used a χ 2 minimization routine to find those parameters that best fit the observed Φ(L) and r (L). Here we seek to constrain two independent CLFs, namely that of the early-type galaxies, hereafter Φ e(l M), and that of the late-type galaxies, hereafter Φ l (L M). In principle we could use the same parameterizations as in Paper 1 for both of these CLFs independently. However, there is an additional constraint that these CLFs have to fulfill: their sum must be equal to the CLF of the entire galaxy population, and thus be consistent with the Φ(L) andr (L) of the combined sample. We therefore use a slightly different, two-step method which automatically obeys the constraint that Φ(L M) =Φ l (L M) +Φ e(l M). We first use the same method as in Paper 1 to obtain the CLF Φ(L M) of the combined sample (step one). Next we introduce the function f l (L, M), which specifies the late-type fraction of galaxies with luminosity L in haloes of mass M. TheCLFs of late- and early-type galaxies are then given by Φ l (L M)dL = f l (L, M)Φ(L M)dL (12) and, by definition, Φ e(l M)dL =[1 f l (L, M)] Φ(L M)dL (13) What remains (step two) is to find the f l (L, M) that,given our best-fit CLF for the combined population, best fits the LFs and correlation lengths of the early- and late-type galaxies. Following Paper 1 we assume that the CLF of the combined sample can be described by a Schechter function: Φ(L M)dL = Φ ( ) α exp( L/ L L L L )dl (14) Here L = L (M), α = α(m) and Φ = Φ (M); i.e., the three parameters that describe the conditional LF depend on M. In what follows we do not explicitly write this mass dependence, but consider it understood that quantities with a tilde are functions of M. We adopt the same parameterizations of these three parameters as in Paper 1, which we repeat here for completeness. Readers interested in the motivations behind these particular choices are referred to Paper 1. For the total massto-light ratio of a halo of mass M we write M (M) = 1 ( M ) [ ( M ) γ1 ( M ) γ2 ] +, (15) L 2 L M c M c which has four free parameters: a characteristic mass M c, for which the mass-to-light ratio is equal to (M/L),and two slopes, γ 1 and γ 2, which specify the behavior of M/L at the low and high mass ends, respectively. A similar parameterization is used for the characteristic luminosity L : M L (M) = 1 ( ) [ ( ) M M γ1 ( ) M γ3 ] f( α) +. (16) 2 L M c M 2 Here f( α) = Γ( α +2) Γ( α +1, 1). (17) with Γ(x) the Gamma function and Γ(a, x) the incomplete Gamma function. This parameterization has two additional free parameters: a characteristic mass M 2 and a power-law slope γ 3.For α(m) we adopt: α(m) =α 15 + ζ log(m 15). (18) Here M 15 is the halo mass in units of 1 15 h 1 M, α 15 = α(m 15 =1),andζ describes the change of the faint-end slope α with halo mass. Note that once α and L are given, the normalization Φ of the CLF is obtained through equation (15), using the fact that the total (average) luminosity in a halo of mass M is given by L (M) = Φ(L M) L dl = Φ L Γ( α +2). (19) Finally, we introduce the mass scale M min below which the CLF is zero; i.e., we assume that no stars form inside haloes with M < M min. Motivated by reionization considerations (see Paper 1 for details) we adopt M min = 1 9 h 1 M throughout. This lower-mass limit does not significantly influence our results. For instance, changing M min to either 1 8 h 1 M or 1 1 h 1 M has only a very modest impact on the results presented below. In order to split the CLF in early- and late-type galaxies we make the assumption that f l (L, M) has a quasi-separable form f l (L, M) =g(l) h(m) q(l, M) (2) Here { 1 if g(l) h(m) 1 q(l, M) = 1 (21) if g(l) h(m) > 1 g(l) h(m) is to ensure that f l (L, M) 1. For the g(l) andh(m) adopted here (see below), q(l, M) = 1 for the vast majority of the relevant (L, M) parameter space, and therefore f l (L, M) can be considered separable to good accuracy. This basically implies that the shapes (but not the absolute values) of the conditionals f l (L M) andf l (M L) are independent of M and L, respectively. Note that g(l) isnotthe same as the fraction F l (L) of all galaxies with luminosity L that are late-type, which instead is given by F l (L) = Φ l(l) Φ(L) = f l (L, M)Φ(L M) n(m)dm (22) Φ(L M) n(m) dm Similarly, the fraction of all galaxies in haloes of mass M that are late-type is given by f l (L, M)Φ(L M)dL F l (M) = (23) Φ(L M)dL The challenge now is to find the g(l) and h(m) that, given the CLF for the combined sample, reproduce the observed LFs and r (L) of the early- and late-type galaxies. To achieve this we use the following estimate for g(l): g(l) = ˆΦ l (L) Φ(L M) n(m) dm (24) ˆΦ(L) Φ(L M) h(m) n(m) dm where ˆΦ l (L) andˆφ(l) correspond to the observed LFs of the late-type and combined galaxy samples, respectively. For any L for which f l (L, M) is separable (i.e., for which q(l, M) = 1 for all M), this implies that the LFs of the early- and late-type galaxies are given by

6 6 van den Bosch, Yang & Mo Φ e,l (L) =ˆΦ e,l (L) Φ(L) ˆΦ(L), (25) i.e., as long as the observed LF of the combined sample is well fit by the model, the same will be true for the LFs of the early- and late-type galaxies. For the L for which q(l, M) < 1 a (typically small) correction to (25) applies. What remains is to find the h(m) that best reproduces the r (L) of the early- and late-type galaxies. This requires the average biases b l and b e for late- and early-type galaxies, respectively, which are obtained from equations (9) and (1) by replacing Φ(L M) withφ l (L M) andφ e(l M), respectively. After experimenting with a variety of different functional forms for h(m) we finally decided to adopt ( [ ( )]) log(m/m) h(m) = max, min 1, (26) log(m 1/M ) Here M and M 1 are two free parameters, defined as the masses at which h(m) takes on the values and 1, respectively. Note that both h(m) andg(l) can in principle take on any positive value. However, for computational reasons we limit h(m) to take values in the range [, 1]. This does not impact our results, as any linear scaling of h(m) reflects itself in g(l) through equation (24). Our model for the three CLFs thus contains a total of 1 free parameters: 4 characteristic masses; M c, M, M 1 and M 2, four parameters that describe the various massdependencies γ 1, γ 2, γ 3 and ζ, and 2 normalizations; one for the mass-to-light ratio, (M/L), and one for the faint-end slope of the CLF, α 15. Although this may seem an awful lot, it is important to realize that we use this model to fit 67 independent data points.furthermore,asweshowinsection 5 below one can actually use observational constraints to fix several of these free parameters. In addition, the data is of sufficient quality to constrain the model freedom. Alternatively, we could have chosen a more restrictive (with fewer free parameters) form for the CLFs, but lacking both observational and physical motivations for a more preferred form of the CLF we felt the need to be sufficiently general. On the other side, one might argue that because of this lack, our model is actually too constrained. For instance, the fact that we assume a Schechter function for the CLF is at best weakly motivated by the observed LFs of clusters of galaxies (e.g., Muriel, Valotto & Lambas 1998; Beijersbergen et al. 22; Trentham & Hodgkin 22; Martínez et al. 22). However, as we show below, our model can accurately fit all observations. Therefore, the use of more general models would only results in a larger amount of degeneracy and will therefore have to await more stringent constraints from either the SDSS or the completed 2dFGRS. 5 RESULTS The cosmological model sets the halo mass function n(m)dm, the dark matter two-point correlation function ξ dm (r), and the halo bias function b(m). Therefore, different cosmologies imply different halo occupancy functions. In Because of our definition of g(l) only 67 of the in total 155 data points are independent. this paper we focus on the ΛCDM cosmology with Ω =.3, Ω Λ =.7, h =.7, Γ = Ω h =.21, and σ 8 =.9. These cosmological parameters are consistent with recent measurements of the cosmic microwave background (e.g., Pryke et al. 21) and large scale structure measurements (e.g., Tegmark, Hamilton & Xu 22). In addition, we have shown in Paper 1 that this cosmology is also the most successful in explaining the LF and galaxy clustering luminosity dependence observed. Therefore, in what follows, we restrict ourselves to this concordance cosmology. Having specified our model, the observational constraints, and the cosmological parameters, we now proceed as follows. We first use Powell s multi-dimensional direction set method (e.g., Press et al. 1992) to find the parameters of the CLF of the combined sample that minimize χ 2 = χ2 (Φ) + χ2 (r ). (27) N Φ N r Here N Φ [ Φ(Li) χ 2 (Φ) = ˆΦ(L ] 2 i), (28) ˆΦ(L i=1 i) and N r [ ] 2 r(l i) χ 2 ˆr (L i) (r )=, (29) ˆr (L i) i=1 with ˆΦ(L i)andˆr (L i) the observed values, and N Φ =44 and N r = 8 the number of corresponding data points. Note that the scaling of χ 2 with N Φ and N r implies that we assign equal weights to χ 2 (Φ) and χ 2 (r ). This differs from the proper statistical χ 2 for which each individual measurement should receive equal weights. However, since N Φ >N r and since the errors on ˆΦ(L) are typically much smaller than the errors on ˆr (L), the χ 2 minimization routine would give much more weight to fitting the LF than to fitting the correlation lengths. Note that the proper χ 2 is only well defined if there are no systematic errors in the measurements and if the error properties of ˆΦ(L) andˆr (L) are the same. It is unlikely that these requirements are fulfilled, and our modified χ 2 is therefore equally meaningful as that of the proper χ 2. We start our investigation using all available model freedom, which means that we minimize χ 2 over a total of 1 free parameters. In the first step we find the 8 parameters of the CLF that best fits the LF and correlation lengths of the combined sample. In the second step, we then find the M and M 1, describing h(m), that best reproduce the observed clustering properties of the early- and late-type galaxies. We refer to this model with maximum freedom as model A. The resulting best-fit parameters are listed in Table 1, and the fits to the data are shown in the upper panels of Figure 2 (solid lines). Overall the fit to the data is remarkably good. The fact that the model reproduces even the small scale features in the LF of the early-type galaxies is a direct consequence of the way in which we have defined g(l) (see Section 4). The lower two panels of Figure 2 plot the number fractions of late-type galaxies as function of luminosity, F l (L), and mass, F l (M). The fraction of late-type galaxies decreases with both increasing luminosity and mass. Note that this is, at least qualitatively, in agreement with the

7 Linking Early and Late Type Galaxies to their Dark Matter Haloes 7 Figure 2. Results for model A. The upper two panels plot the observed LFs (left panel) and correlation lengths (right panel). Symbols are the same as in Figure 1, while the solid lines indicate the model results. For clarity, the LFs have been separated from each other by one order of magnitude in the y-direction, with the combined LF untranslated. Similarly, the r (L) of the early- and late-type galaxies are offset by +2h 1 Mpc and 2h 1 Mpc, respectively. Note the good quality of the fits. The lower panels plot the fraction of late-type galaxies as function of luminosity (left) and halo mass (right).

8 8 van den Bosch, Yang & Mo Figure 3. A comparison of the five models discussed in the text. The upper panel on the left plots the correlation lengths as function of luminosity. Symbols with errorbars correspond to the 2dFGRS data with the r (L) of the late-type and early-type galaxies translated by 2h 1 Mpc and +2h 1 Mpc, respectively. The fits of these models to the observed LFs are not shown, since the differences between the various models can not be discerned by eye. The other three panels plot, for all five models, the inferred mass-to-light ratios M/L (M) (upper right), the late-type fractions F l (M) (lower left) and the average total number of galaxies N tot(m) (lower right). Model E predicts mass-to-light ratios and late-type fractions for cluster-sized haloes that are inconsistent with data. In order to discriminate between the other models, better constraints on r for faint, early-type galaxies and/or accurate, independent measurements of F l (M) and/or M/L for haloes with M > 114 h 1 M are required.

9 Linking Early and Late Type Galaxies to their Dark Matter Haloes 9 Table 1. Model parameters. ID logm L (M/L) cl logm c logm logm 1 logm 2 (M/L) γ 1 γ 2 γ 3 ζ α 15 χ 2 (Φ) χ 2 (r ) (1) (2) (3) (4) (5) (6) (7) (8) (9) (1) (11) (12) (13) (14) (15) A B C D E B B Column (1) lists the ID by which we refer to each model in the text. Columns (2) to (13) list the best-fit model parameters, where parameters that were kept fixed during the fitting procedure are type-set in boldface. Here M L is defined such that L (M L)=L (see Section 5), (M/L) cl is the mass-to-light ratio of haloes with M 1 14 h 1 M,andα 15 is the faint-end slope of the conditional LF for haloes with M =1 15 h 1 M. Columns (14) and (15) list the values of χ 2 (Φ) and χ 2 (r )of the best-fit model, respectively. Here χ 2 (Φ) corresponds to the χ 2 of the fit to the LF of the combined sample only (N Φ = 44), whereas χ 2 (r ) is summed over all r measurements of all three samples (combined, early- and late-type; N r = 23). Masses and mass-to-light ratios are in h 1 M and h (M/L), respectively. Figure 4. Contour plot of χ 2 (r ) as function of M and M 1 for model B. Contours correspond to χ 2 (r )=5.5, 6.5, 7.5,..., The location of the best-fit model is indicated by a thick dot labeled B. The locations of models B1 and B2, discussed in the text, are also indicated. The dashed, diagonal line corresponds to M = M 1 and indicates the boundary above which no good fits can be obtained. For models with M 1 <M there is a large area in (M,M 1 ) parameter space that yields virtually identical χ 2 (r ). well-known morphology-density relation. The noisy wiggles in F l (L) are due to the fact that g(l) is computed directly from the observed LFs, while the feature in F l (M) for M < 3 11 h 1 M is a consequence of the upturn in the LF of early-type galaxies at L < 18 h 2 L. The upper-right panel of Figure 3 plots the total, average mass-to-light ratio of model A as function of halo mass, computed using equation (2). The M/L (M) reveals a sharp minimum of 11h(M/L) at M h 1 M. For M< h 1 M the mass-to-light ratios increase dramatically with decreasing mass, such that in haloes with M < 11 h 1 M virtually no (blue) light is produced. This sharp increase of M/L with decreasing halo mass is required in order to bring the steep slope of the halo mass function at low M in agreement with the relatively shallow faint-end slope of the observed LF. In the language of galaxy formation models; low-mass haloes need efficient feedback to prevent an overabundance of faint galaxies. For haloes with M> h 1 M the (average) mass-to-light ratio increases roughly as M/L M.3, such that clusters of 1 15 h 1 M have, on average, a (blue) mass-to-light ratio of 1h (M/L). Within the context of galaxy formation models, this increase in mass-to-light ratio is interpreted as a decrease of the cooling efficiency in more massive haloes (e.g., White & Rees 1978; White & Frenk 1991). By construction, M/L M γ 2 for haloes with M M c. However, several studies have suggested that on cluster mass scales M/L varies only weakly with mass (e.g., Bahcall, Lubin & Norman 1995; Bahcall et al. 2; Kochanek et al. 22). Furthermore, Fukugita, Hogan & Peebles (1998), using a variety of observational constraints, derived that clusters of galaxies have blue mass-to-light ratios of 45 ± 1h (M/L), i.e., more than 5σ lower than what model A predicts for a Coma sized cluster. Therefore, we now consider a model (model B) in which we keep M/L fixed at a constant value of (M/L) cl for haloes with M 1 14 h 1 M. Motivated by Fukugita et al. (1998) we adopt (M/L) cl = 5h (M/L).Continuityof M/L (M) then sets the parameter γ 2 which is therefore no longer a free parameter. With respect to model A, the minimum of M/L (M) occurs at a somewhat lower mass of 1 11 h 1 M,thefraction of late-type galaxies in massive haloes has increased, and the fit to the correlation lengths has improved, especially for the faint early-type galaxies (see Figure 3). This is easy to understand: Lowering the mass-to-light ratios on cluster scales means that more galaxies have to reside in clusters. Since more massive haloes are more strongly clustered

10 1 van den Bosch, Yang & Mo (Mo & White 1996) and the majority of cluster galaxies are early-types, this leads to an increase of r which is most pronounced for the early-types. Although overall model B is in better agreement with the data than model A, it also has an unattractive feature. This is illustrated in the lower right panel of Figure 3 where we plot the average, total number of galaxies N tot(m) = Φ(L M)dL = Φ Γ( α + 1) (3) as function of halo mass. Model B predicts that this number is less than unity for haloes with masses around 1 11 h 1 M. Yet, this mass scale coincides with the minimum of M/L (M). This implies that these haloes must have an extremely large spread in M/L; the majority of haloes contain zero galaxies (or only dark galaxies, which produce no light), while a small fraction harbors one (or more) relatively bright galaxy (in order to explain the M/L ). This aspect of the model is due to the fact that theclfforhaloesofthismassscalehasarelativelyhigh characteristic luminosity L. If we define M L as the mass for which L (M) =L,whereL = h 2 L is the characteristic luminosity of the observed LF of the combined sample (see Madgwick et al. 22), we find for model B a relatively low value of M L 1 13 h 1 M (see Table 1). Since the abundance of haloes with this mass is relatively high, reproducing the observed abundance of L galaxies requires a significant fraction of haloes with dark or no galaxies. Although the existence of dark galaxies is interesting in itself (see Verde, Oh & Jimenez 22), a large scatter in M/L for haloes with M 1 11 h 1 M seems inconsistent with the small scatter in the observed Tully-Fisher and Fundamental Plane relations (see also discussion in Paper 1). Therefore, in models C and E we tune the value of M 2 such that M L = h 1 M and h 1 M, respectively (i.e., M 2 is not a free parameter here). We again set M/L constant for haloes with M 1 14 h 1 M, but we let the value of (M/L) cl be a free parameter. These models thus have the same number of free parameters as model B. As can be seen from Figure 3, model C is remarkably similar to model A. This is mainly due to the fact that the value of M L of model A is with close to that of model C (see Table 1). The best-fit value of (M/L) cl is with 615h (M/L) consistent with the value advocated by Fukugita et al. (1998) at the 2σ level. Model E, however, is dramatically different. The minimum of M/L (M) occurs at a significantly higher mass scale of h 1 M. In addition, (M/L) cl 1h (M/L) and F l = for haloes with M > 115 h 1 M, both of which are inconsistent with observations. Furthermore, model E yields a significantly poorer fit to the observed correlation lengths of the faint early-type galaxies, and we therefore no longer consider model E in what follows. Finally, in model D, we combine the constraints from models B and C above; model D has therefore only 8 free parameters together with M L = h 1 M and (M/L) cl = 5h (M/L). With respect to model C this causes a small increase in correlation lengths (again because more light is added to the strongly clustered massive haloes), though the effect is sufficiently small that model D can still be considered as yielding an overall good fit. Figure 4 plots contours of constant χ 2 (r )inthem versus M 1 plane for model B (results for the other models discussed above are very similar). The dashed diagonal line corresponds to M = M 1 (i.e., h(m) is a step-function). Clearly, models with M <M 1 (for which the fraction of late-type galaxies increases with halo mass) result in poor fits as the value of χ 2 (r ) is always extremely large. This is easy to understand. Since the correlation lengths of the late-type galaxies are smaller than those of the early-type galaxies, and since more massive haloes are more strongly clustered, reproducing the observed r (L) requires that the late-type fraction decreases with halo mass, and thus that M >M 1. However, as is also apparent from the contour plot, a large area in the M -M 1 plane yields roughly equally good fits. To illustrate this degeneracy we have also computed two additional models, B1 and B2, for which we keep all parameters identical to that of model B, but we modify M and M 1 so that these models fall in the region of roughly the same χ 2 (r ) (indicated by solid dots in Figure 4). Note that changes in h(m) only affect the r (L) oftheearlyand late-type galaxies; all LFs and the correlation lengths of the combined sample are unaffected. The results for models B1 and B2 are shown in Figure 5. As is also apparent from their values of χ 2 (r ) listed in Table 1, these models are virtually indistinguishable from model B and from each other. However, they do yield somewhat different F l (M). Model B2 for instance yields late-type fractions of zero for M > 115 h 1 M. This is inconsistent with data, which allows us to rule against this particular model. However, models B1 and B have quite similar F l (M), and current data is not sufficient to discriminate between these two alternatives. This degeneracy also explains why we have adopted a fairly simple form for h(m); more complicated forms, with more free parameters, only increases the amount of degeneracy. Having shown that there are different models that fit the data roughly equally well, the question arises whether one can discriminate between these different halo occupancy models. Based on the observed mass-to-light ratios of clusters, and on the amount of scatter inferred from the Tully- Fisher and Fundamental Plane relations, one might argue that of the models presented above, model C is to be preferred (modulo some uncertainties in M and M 1). In what follows, however, we always compare predictions from different models. This gives an idea about the extent of the uncertainties in the models, and the accuracy in the data required to allow to discriminate between the various models. Some clues are already apparent from Figure 3. As we have shown above, the models are extremely sensitive to the exact value of M L and most of the uncertainties in the models canbetranslatedtovaluesofm L in the range 1 13 h 1 M to h 1 M. In order to further constrain this crucial parameter more accurate measurements are required of the mass-to-light ratios and late-type fractions in clusters or of the correlation lengths of galaxies with M bj 5logh > 18. In summary, within the concordance cosmology several models can be found that fit the data remarkably well. Although some amount of degeneracy exists, the mean trends reveal an average mass-to-light which has a minimum of 1 h (M/L) at around h 1 M,and which increases extremely rapidly towards lower masses: in all cases we find M/L > 3 h (M/L) for haloes with M<1 1 h 1 M. The correlation lengths require that the fraction of late-type galaxies decreases from about 9 per-

11 Linking Early and Late Type Galaxies to their Dark Matter Haloes 11 Figure 5. Correlation lengths and late-type fractions as function of magnitude and halo mass, respectively, for models B, B1, and B2. Although these three models yield virtually indistinguishable r (L), they imply F l (M) that are quite different. Note that model B2 can be rejected based on its prediction of zero late-type galaxies in haloes with M>1 15 h 1 M. cent for haloes with M < 112 h 1 M to anywhere between and 4 percent on scales of M =1 15 h 1 M, in qualitative agreement with the morphology-density relation. Similarly, the observed LFs indicate a similar trend with luminosity: about 9 percent of all galaxies with L < 19 h 2 L are latetypes, while this drops to about 2-3 percent for galaxies brighter than L. 6 TWO POINT CORRELATION FUNCTIONS Two point correlation functions are a powerful tool to describe the clustering properties of galaxies. With the CLFs for early- and late-type galaxies we can compute the galaxygalaxy correlation function, ξ gg(r), not only as function of galaxy type, but also as function of luminosity. This allows a detailed investigation of the bias of galaxies as function of scale, luminosity, and (spectral) type. At small scales, where the number of pairs are mostly due to galaxies within the same halo, ξ gg(r) depends not only on the occupation numbers of galaxies (which can be obtained from the CLF), but also on how galaxies are distributed inside individual dark matter haloes. In addition, since the number of pairs within a halo containing N galaxies is equal to 1 N(N 1) one also needs to know the second 2 moment of the halo occupation number distribution. Since the CLF only contains information about the first moments N(M) we need to make additional assumptions. Alternatively, as shown by Cooray (22), one may use the observed power spectrum of galaxies to obtain constraints on the second moments of P (N M) using standard inversion techniques. We follow Yang et al. (22) and make the assumption that the probability distribution P (N M), with N an integer, is given by P (N M) = { N +1 N(M) if N = N N(M) N if N = N +1 otherwise (31) Here N is the integer of N(M). Thus, the actual, integer number of galaxies in a halo of mass M is either N or N +1. This particular model for the distribution of halo occupation numbers is supported by the semi-analytical models of Benson et al. (2) who found that the halo occupation probability distribution is narrower than a Poisson distribution. In addition, they have shown that distribution (31), which they call the average distribution, is successful in yielding power-law correlation functions, much more so than for example a Poisson distribution (see also Berlind & Weinberg 22). For this average distribution the average number of galaxy pairs inside an individual dark matter halo is given by N pair(m) = N N(M) 1 2 N (N + 1) (32) For the spatial distribution of galaxies within a halo we follow Peacock & Smith (2), Benson et al. (2) and Berlind & Weinberg (22) and assume that (i) the brightest galaxy resides in the center, and (ii) the remaining satellite galaxies follow the density distribution of the dark matter (which we specify in Appendix A). The special treatment of a central galaxy is required if the galaxy-galaxy correlation function is to remain close to a single power-law, rather than to reveal a flattening as present in the dark matter mass correlation function (Peacock & Smith 2; Berlind & Weinberg 22). Pairs between the central galaxy and satellite galaxies follow a separation function f cs(r) =4π ρ(r) r 2,with ρ(r) the normalized density distribution of the halo, which integrates to a total halo mass of unity. Satellite-satellite pairs follow a different separation function f ss(r) whichcan also be obtained from the halo density distribution. The sep-

12 12 van den Bosch, Yang & Mo Figure 6. Real-space correlation functions of all galaxies (solid lines), late-type galaxies (dotted lines), and early-type galaxies (dashed lines). Results are shown for four different models, and for four different magnitude bins (as indicated). Note that M b is defined as J M bj 5logh. The dot-dashed curves in each panel correspond to the real-space correlation function of the evolved, non-linear dark matter mass distribution, and is plotted for comparison. Note that the ratio between the correlation functions of early- and late-type galaxies increases with decreasing r. See text for a detailed discussion.

13 Linking Early and Late Type Galaxies to their Dark Matter Haloes 13 Figure 7. Same as Figure 6, except that here we plot the projected correlation function w p(r p)/r p instead of the real-space correlation function ξ(r). Note that these projected correlation functions, which unlike the real-space correlation functions can be directly obtained from redshift surveys, are much smoother than the real-space correlation functions and much better resemble single power-laws.

14 14 van den Bosch, Yang & Mo aration function for all pairs, f(r), can be written as N pair(m) f(r)dr = N cs(m) f cs(r)dr + N ss(m) f ss(r)dr (33) where N cs(m) and N ss(m) correspond to the average number of central-satellite and satellite-satellite pairs, respectively. Once the CLF and N pair(m) f(r) are specified, ξ gg(r) can be computed (see Appendix A). Figure 6 plots ξ gg(r) for the late-type galaxies (dotted lines), the early-type galaxies (dashed lines) and the combined sample of galaxies (solid lines) for different luminosity bins and models. In addition, we plot the dark matter mass correlation function (dotdashed lines) for comparison. Panels in the upper three rows show results for galaxies in magnitude intervals for which data from the 2dFGRS exists (Norberg et al. 22). The panel in the lower row corresponds to 14 >M bj 5logh > 18. Since Norberg et al. only presented correlation functions for galaxies with M bj 5logh 17.5, no data on ξ gg(r) exists for this magnitude range. Virtually all correlation functions, including that of the dark matter, reveal a special feature in the radial interval 1h 1 Mpc < r < 3h 1 Mpc. This coincides with the transition from the 1-halo term to the 2-halo term (see Appendix A): at r < 1h 1 Mpc the correlation functions are governed by galaxy pairs within individual dark matter haloes and therefore depend on our assumptions regarding N pair(m) f(r). For r > 3h 1 Mpc, however, the galaxy correlation functions are independent of how galaxies are distributed inside haloes and only depend on the first moments N(M) of the halo occupation number distributions. Typically, at r < 2h 1 Mpc the ratio of the correlation functions of the early- and late-type galaxies is larger than at r > 2h 1 Mpc. Since at small scales the clustering strength depends on the second moment of the occupancy numbers, the correlation function at small r is more dominated by the contribution from massive haloes than at large r. The ratio between the ξ gg(r) of early- and late-type galaxies at small r is therefore sensitive to F l (M) at large M. This is immediately apparent from a comparison with the lower left panel of Figure 3, which shows that models A and C are very similar with lower late-type fractions than models B and D. Consequently, the ratio of ξ gg(r) of early- and latetype galaxies at small r is larger for models A and C than for models B and D. As is also immediately apparent from Figure 6, the bias of galaxies with respect to the dark matter mass distribution depends sensitively on galaxy luminosity, on the radial scale, and on galaxy type (see also Kauffmann, Nusser & Steinmetz 1997). In general, early-type galaxies are biased tracers of the mass distribution at all scales, and for all luminosities, whereas late-type galaxies are virtually always anti-biased. The combined sample of early- and late-type galaxies reveals a transition from positive bias at large scales (r > 2h 1 Mpc) to anti-bias at smaller scales, except for the most luminous galaxies which are always positively biased. 6.1 Comparison with observations Can we compare these ξ gg(r) to those obtained from the 2dFGRS? Norberg et al. (22) argued that the real-space correlation functions of all subsamples of the 2dFGRS data are well fit by a single power-law (independent of luminosity or spectral type). Taken at face value, the fact that virtually all correlation functions shown in Figure 6 reveal clear deviations from pure power-laws would force us to rule against each of our models. However, it is important to take into account how real-space correlation functions are obtained from data. After all, the data only yields a redshiftspace correlation function, which is distorted with respect to the real-space correlation function due to the peculiar motions of the galaxies. On small scales the virialized motion of galaxies within dark matter haloes cause a reduction of the correlation power (the so-called finger-of-god effect), while on larger scales the correlations are boosted due to coherent flows (Kaiser 1987). It is common practice to therefore compute the galaxy correlation function on a twodimensional grid of pair separations parallel (π) and perpendicular (r p) to the line-of-sight. Integration of this ξ gg(r p,π) over π then yields what is called the projected correlation function, which is related to the real-space correlation function by an Abel transform w p(r p)= ξ gg(r p,π)dπ =2 r p ξ gg(r) r dr r2 r 2 p (34) (Davis & Peebles 1983). If the real-space correlation function is a single power-law, ξ gg(r) =(r/r ) γ then the projected correlation function also follows a single power-law w p(r p)/r p = A(γ)(r p/r ) γ with Γ(1/2) Γ( (γ +1)/2) A(γ) = (35) Γ( γ/2) (see e.g., Baugh 1996). In Figure 7 we plot w p(r p)/r p (obtained using equation [34]) for the same samples of galaxies as in Figure 6. As can be seen, the projection has largely washed-out the features at 2h 1 Mpc in the real-space correlation functions, and overall the projected correlation functions better resemble single power-laws. Only the correlation functions of early-type galaxies with 14 >M bj 5logh > 18, for which no data exists to date, clearly deviates from a single power-law. In order to compare our models to the 2dFGRS data in a more quantitative way, we proceed as follows. For each spectral type and each luminosity bin Norberg et al. (22) have fit a single power-law relation of the form w p(r p)/r p = A(γ)(r p/r ) γ to the projected correlation functions obtained from the 2dFGRS data. We rewrite this as log[w p(r p)/r p]=w + γlog(r p) (36) and compute w = loga(γ) γ logr and its error from the values of r and γ (and their errors) quoted by Norberg et al. (22). The results are shown in Figure 8 (open circles with errorbars). Next, for each of our models we fit equation (36) to the projected correlation functions over the same radial interval over which Norberg et al. (22) fitted the data (.5 log(r p) 1.3). Our best-fit w and γ are shown in Figure 8 for all models listed in Table 1. Overall, the models fit the data reasonably well. Although the models predict somewhat more pronounced dependencies of γ and w on luminosity, and somewhat shallower correlation functions for the late-type galaxies, the errorbars on the data are too large to rule against any